3 Redshift distribution of galaxies in the MS 1008-1224 field

3.1 Photometric redshift estimation

Photometric redshifts (hereafter $z_{\rm phot}$) were computed using the standard fitting procedure hyperz (Bolzonella et al. 2000) which compares the observed spectral energy distribution (SED) of a given galaxy, obtained from photometry, to a set of template spectra. Redshifts are then computed using a standard $\chi^2$ minimization. hyperz explores the parameter space defined by the age and metallicity of the stellar population, the IMF, the reddening law and the reddening value. When tested against the HDF spectroscopic sample $z_{\rm phot}$ errors from hyperz were found to be $\Delta z \sim 0.05$ at $z \le 1$, and $\Delta z \sim 0.1 (1+z)$ for more distant galaxies (see Bolzonella et al. 2000 for more details).

For this work, we used a set of 8 template families from the new Bruzual & Charlot evolutionary code (GISSEL98; Bruzual & Charlot 1993) with a Miller & Scalo IMF. The families spanned a wide range of ages for the stellar population and included: a single burst (coeval stellar population), a constant star-forming rate, and six $\mu$-models (exponentially decaying SFR) designed to match the sequence of colours from E-S0 to Sd galaxies (255 spectra in all). The reddening law was taken from Calzetti et al. (2000) with values of A_V between 0 and 1.2 mag, the upper value being twice the mean E(B-V) reported by Steidel et al. (1999) for galaxies up to $z \sim 4$. The Lyman forest blanketing was modelled according to Madau (1995). hyperz computes error bars corresponding to 69, 90 and 99 per cent confidence levels computed by means of the $\Delta\chi^2$ increment for a single parameter (Avni 1976). We only considered a $z_{\rm phot}$ estimate when the best fit template had $\chi^2 <$ 1. It may be noted that errors in the photometry of the MS 1008-1224 field were more significant than uncertainties in the template spectra used.

The accuracy and robustness of $z_{\rm phot}$ were investigated using simulated catalogues of galaxies with realitics SEDs. The error budget and $z_{\rm phot}$ accuracy were then analysed as a function of the ESO-BVRIJK filter set, the photometric errors and the redshift range of the simulated galaxies.

First, catalogues were produced for the two sets of filters, BVRI (FORS1 images alone) and BVRIJK (field common to FORS1 and ISAAC images), assuming a uniform redshift distribution and a Gaussian photometric error distribution of fixed sigma (0.1 mag), uncorrelated between the different filters. These were used to determine (i) $z_{\rm phot}$ errors, (ii) the fraction of sources for which hyperz returned either no solution ( $\chi^2 > 1$) or multiple solutions and (iii) the fraction of sources with spurious values (i.e., errors much larger than the normal dispersion at that redshift). The uniform distribution of simulated galaxies across the redshift range provided a sufficient number of objects for a robust estimation of the errors at all redshifts.

We then performed a second set of simulations using a pure luminosity evolution (PLE) model. The redshift distribution and the photometric errors (a function of magnitude and filter) in this second simulation were tailored to mimic the VLT observations of the MS 1008-1224 field in a more realistic manner. In particular, we focussed on (simulated) galaxies in the range $22.5 \le R \le 26.5$ (shear analysis sample) and $22.5 \le I \le 25.5$(depletion analysis sample). Galaxies were assigned magnitudes and colours randomly according to the PLE model of Pozzetti et al. (1998) which had been designed to reproduce the deep B counts (Williams et al. 1996).

The results from the simulations are shown in Fig. 2 (uniform redshift distribution) and Fig. 3 (PLE distribution).

Figure 2: Photometric versus model redshift for a simulated catalogue of galaxies uniformly distributed in redshift and a $10\%$ photometric error. Error bars of $\pm 1\sigma$ are shown for each object. The 5 diagonal lines plotted are $z_{\rm phot}= z({\rm model})$, $z_{\rm phot} = z({\rm model}) \pm 0.05$ and $z_{\rm phot} = z({\rm model}) \pm 0.1$. Top panel: $z_{\rm phot}$ computed using BVRIJK photometry. Bottom panel: $z_{\rm phot}$ computed using BVRI photometry.

Figure 3: Photometric versus model redshift for a simulated catalogue of galaxies simulated to match the data in hand, with a PLE number counts model and photometric errors scaling with magnitude. Error bars of $\pm 1\sigma$ are shown for each object. The 5 diagonal lines plotted are $z_{\rm phot}= z({\rm model})$, $z_{\rm phot} = z({\rm model}) \pm 0.05$ and $z_{\rm phot} = z({\rm model}) \pm 0.1$. Top panel: $z_{\rm phot}$ computed using BVRIJK photometry. Bottom panel: $z_{\rm phot}$ computed using BVRI photometry.

It is clear from these plots that most galaxies with BVRIJK magnitudes will have fairly well determined $z_{\rm phot}$. Typical uncertainties scale as $\Delta z_{1\sigma} \sim 0.1 (1+z)$, and the fraction of objects with no solution or spurious values is only a few per cent. Therefore we are confident that the redshift distribution of galaxies on the ISAAC field has been well determined. In contrast, the situation is less satifying when using only BVRI in the redshift range $0.8 \le z \le 2.5$. This is due to the lack of strong spectral features in the wavelengths covered by these filters. The main problem in this case is the translocation of some fraction of $z \ge 1$ objects into lower redshift bins.

The redshift ( $z_{\rm phot}$) distribution of sources in the ISAAC field is shown in Fig. 4.

Figure 4: Photometric redshift distribution of galaxies inside the ISAAC field obtained from BVRIJK data. The peak corresponds to the cluster MS 1088-1224. Only those galaxies which fit the hyperz models with $\chi ^2 < 1$ are plotted here (see Sect. 3). The inset histogram represents galaxies with $22.5 \le I \le 25.5$ and $z_{\rm phot} \ge 0.4$ (i.e. well behind the cluster).

This distribution comprises galaxies for which hyperz returned a unique $z_{\rm phot}$ value with a good model fit ( $\chi ^2 < 1$). To satisfy this criterion we excluded blended sources from the analysis.

The cluster of galaxies comprising MS 1008-1224 is an "in-situ'' control sample for checking the accuracy of our $z_{\rm phot}$ estimation. Indeed, the cluster shows up as a prominent spike in the $z_{\rm phot}$ distribution (Fig. 4) between z = 0.25 and 0.4 ( $z_{\rm mean} \sim 0.34$) which confirms both the efficacy of the method as well as our error estimates from simulations. An additional, and unexpected, check was provided by the discovery of a background cluster at $z_{\rm phot} \sim 0.9$. That a number of galaxies were clustered in redshift space as well as on the sky suggested that their $z_{\rm phot}$ value was reasonably accurate.

   
3.2 Distribution of cluster galaxies

We also simulated cluster fields at z = 0.31 as targets for the hyperz program. The clusters were generated with galaxies distributed according to a King model with central line-of-sight velocity dispersion of 1000 kms^-1, a core radius of 500 kpc and a Schecter luminosity distribution in the range $-14 \le M_B \le -22$. The mixture included 70 per cent ellipticals and S0 galaxies, 28 per cent spirals and 2 per cent star-forming galaxies. The other parameters (IMF, SFR, models etc.) were as described before. The photometric accuracy (as a function of magnitude) and limiting magnitudes were chosen to match the observed values for MS 1008-1224. The apparent magnitudes in all filters were computed through GISSEL98. This simulated cluster catalogue was added to a PLE field catalogue to simulate the observed catalogue.

Figure 5 shows the cluster sequence on the observed Colour-Magnitude plot R-I vs. R.

Figure 5: Top: a Colour-Magnitude (R-I vs. R) plot for the field of MS 1008-1224. The cluster sequence formed by galaxies belonging to MS 1008-1224 is clearly visible as a horizontal strip at $R{-}I = 0.69 \pm 0.15$. Bottom: colour-magnitude plot for a simulated cluster field. Cluster and field galaxies are displayed as black dots and open circles respectively.

It is almost horizontal for these filters. The same Colour-Magnitude diagram is also plotted for a realization of the simulated cluster described above and is very similar to the actual data.

Simulations indicated that the error in $z_{\rm phot}$ for cluster members was $\sim$0.04 ($1\sigma$) which is similar to the width of the peak obtained for real data (Fig. 4). It may be noted that this accuracy is much better than $\Delta z_{1\sigma} \sim 0.1 (1+z)$ because of the presence of appropriately located spectral features which make identification of red cluster galaxies particularly easy in this redshift range. It is also clear that the cluster redshift distribution is skewed, the lower redshift side of the peak being considerably steeper than its counterpart. So we defined the foreground galaxy sample as those at $z_{\rm phot} < 0.25$ and the background sample (for lensing analysis) as those at (conservatively) $z_{\rm phot} > 0.4$. hyperz detected 75 per cent of the simulated cluster galaxies in the range $0.25 \le z_{\rm model} \le 0.35$ with BVRIJK.

One of the problems with using the BVRI photometry for $z_{\rm phot}$ is the contamination of lower redshift bins by interlopers from z > 1 and this could be higher than 1 in 3 sources. On the contrary, the simulated Colour-Magnitude diagram in Fig. 4 indicated that the cluster luminosity is dominated (80-90 per cent) by emission from red ellipticals on the cluster sequence. Carlberg et al. (1996) estimated that the red galaxies on the cluster sequence underestimate the cluster luminosity by about 15 per cent, similar to what we see in our simulated data. So we used galaxies from the entire FORS1 field on the cluster sequence of the Colour-Magnitude diagram ( $R{-}I = 0.69 \pm 0.15$, 17.5 < R < 24.0) for calculating the luminosity distribution and applied the $15\%$ correction determined above.

The number and luminosity density distributions of cluster galaxies are shown in Fig. 6.

Figure 6: Galaxy number and R-band luminosity density distribution in the galaxy cluster MS 1008-1224. The densities were computed from galaxies on the cluster sequence on the Colour-Magnitude plot (see Fig. 5 and the text) and smoothed with a 50'' Gaussian. The average galaxy number density is 5.9 gal arcmin-1. The number density contours plotted (left) are 12.0, 24.0, 36.0, 72.0, 144.0, 215.0, 280.0, 360.0, 720.0, 1080.0, 1440.0, and 1800.0 gal arcmin-2. So, the density contrast reaches 305 in the cluster center with respect to the average value. The average luminosity over the field is $0.0021 \times 10^{13}~h^{-1}~L_{\odot}$ arcmin-2. The luminosity density contours plotted (right) are 0.0015, 0.003, 0.005, 0.01, 0.02, 0.07, 0.12, 0.2, 0.3, 0.5 and $0.7 \times 10^{13}~h^{-2}~L_{\odot}$ arcmin-2. So the luminosity contrast at the cluster center is 330.

The distributions were obtained by computing the local galaxy number density at each galaxy position followed by a smoothing of the resulting density field by a Gaussian of 50 arcsec FWHM. There was no significant change when the cluster magnitude limit was extended to R = 27. The galaxy numbers and luminosity show a strong concentration around the cD and a northward extension. One can discern four prominent peaks in both distributions at ( $x_{\rm arcsec}$, $y_{\rm arcsec}) \equiv ($220, 220), corresponding to the location of the cD galaxy, (200, 280), (240, 380), (260, 300) and (360, 160). The coherent substructures indicated by the coincidence of light and number peaks suggests that MS 1008-1224 is not yet dynamically relaxed.

3.3 The redshift distribution and lensing analysis

We used photometric redshifts to calculate the average lensing distance modulus to convert the gravitational convergence (shear analysis) into a mass estimate. Thus missing the $z \geq 1$ objects from BVRI photometry would have resulted in a considerable error in the absolute mass estimate. Therefore we assumed that the redshift distribution in the ISAAC field (i.e. those sources which had BVRIJK magnitudes) was representative of the whole FORS1 field.

The redshift of background sources affects the mass estimate only through the angular scale distance which has a weak redshift dependence at z > 0.5 in most cosmologies. So $\delta z_{\rm phot} = 0.1 {-} 0.2$ is not of much concern, especially when compared to other sources of error discussed in the section on shear analysis. We also used the $z_{\rm phot}$ distribution to determine the foreground source contamination of the lensing sample which would have diluted the lensing signal and corrected for the same.